
where \(\sigma =\pm 1\)  and \(g\)  is standard normal independent of the sequence \(\lbrace g_i\rbrace \) . If we denote by \(\mathbb {E}_g\)  the expectation on the probability space \((\Omega _g,\Sigma _g,\mathbb {P}_g)\)  on which the sequence \(\lbrace g_i\rbrace \)  and \(g\)  are defined, then by [1]}, [2]}, it holds that
\(\mathbb {E}_g\bigg [\sup \lbrace Z_1(f,\sigma ): f\in \mathcal {F},\sigma =\pm 1\rbrace \bigg ]\le \mathbb {E}_g\bigg [\sup \lbrace Z_2(f,\sigma ): f\in \mathcal {F}, \sigma =\pm 1\rbrace \bigg ] .\) 
